Question: What is the inverse of the function $g(x)=\dfrac{7x+3}{x-5}$ ? $ g^{-1}(x) =$
Answer: Let's start by replacing $g(x)$ with $y$. $y=\dfrac{7x+3}{x-5}$ Now let's swap $x$ and $y$ and solve for $y$. $\dfrac{7y+3}{y-5}=x$ [Why do we swap x and y?] $\begin{aligned} \dfrac{7y+3}{y-5}&=x \\\\ 7y+3&=x(y-5) \\\\ 7y+3&=xy-5x \\\\ 7y-xy&=-5x-3 \\\\ y(7-x)&=-5x-3 \\\\ y&=\dfrac{-5x-3}{7-x} \end{aligned}$ In conclusion, this is the inverse function: $g^{-1}(x)=\dfrac{-5x-3}{7-x}$ [I saw someone solve this problem by originally solving for x. Were they wrong?]